Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2205.13782 · Physical Review A

Computational characteristics of the random-field Ising model with long-range interaction

Fangxuan Liu, L.-M. Duan

Published Jul 11, 2023• 3 citations
arXivHTMLPDF

Abstract

The Ising model is a widely studied class of models in quantum computation. In this paper, we investigate the computational characteristics of the random-field Ising model (RFIM) with long-range interactions that decays as an inverse polynomial of distance, which can be achieved in the current ion trap system. We prove that for an RFIM with long-range interaction embedded on a two-dimensional plane, solving its ground state is NP-complete for all diminishing exponents, and prove that the one-dimensional RFIM with long-range interaction can be efficiently approximated when the interaction decays fast enough.

DOI
10.1103/physreva.108.012415

Related papers

Critical behaviour of the random-field Ising model with long-range interactions in one dimension

Ivan Balog, Gilles Tarjus, Matthieu Tissier

We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.

arXiv:1407.4891•Journal of Statistical Mechanics Theory and Experiment•Oct 9, 2014•16 citations

Critical behavior of the random-field Ising magnet with long-range correlated disorder

Björn Ahrens, Alexander K. Hartmann

We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ${r}^{a}$, where $r$ is the distance between two lattice sites and $a<0$. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.

arXiv:1106.6244•Physical Review B•Oct 7, 2011•19 citations

Dynamics of the Random Ising Model with Long-Range Interaction

Chen, Yuan, Li +7 more

Critical dynamics of the random Ising model with long-range interaction decaying as r-(d+σ) where d is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model A dynamics. Asymptotic scaling laws are studied in a frame of the expansion in = 2σ - d. In dimensions d ＜ 2σ. the dynamic exponent z is calculated to the second order in at the random fixed point.``

理论物理通讯：英文版•Jan 1, 2001•0 citations

Dynamics of the Random Ising Model with Long-Range Interaction

Yuan Chen, Zhibing Li, Fang Hai +2 more

Critical dynamics of the random Ising model with long-range interaction decaying as (where is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model dynamics. Asymptotic scaling laws are studied in a frame of the expansion in . In dimensions , the dynamic exponent is calculated to the second order in at the random fixed point.

Communications in Theoretical Physics•Oct 15, 2001•1 citation

Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator

Guido Pagano, Aniruddha Bapat, Patrick Becker +13 more

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

arXiv:1906.02700•Proceedings of the National Academy of Sciences•Oct 6, 2020•261 citations

Computational studies of classical disordered spin systems

Yong Wu

In this thesis classical disordered spin systems, in particular, the random field Ising model (RFIM), are studied using numerical methods. The previous experimental and theoretical research on the RFIM is reviewed. Critical exponents of the RFIM are extracted at zero temperature. The heat capacity exponent α is found to be near zero. The ground states within a certain region with varying external field and disorder are portrayed, and it is speculated that if related quantities are scaled properly, ground state portraits at different system sizes are statistically similar. The performance of the Wang-Landau (WL) algorithm, which is used in our positive temperature study of the RFIM, is measured by the mean first-passage time. It is proved that the mean first, passage time of the mean field Ising model scales as N 2 log N, where N is the number of spins. The logarithmic correction term to the unbiased Markovian random walk is due to the boundaries. When applied to the Ising model with nearest neighbor interactions, the WL algorithm suffers from power law slowing down. Carefully designed cluster updating, however, can accelerate the WL algorithm and eliminate the slowing down. The WL algorithm is used in our finite temperature study of the RFIM. Physical quantities, such as the specific heat and the susceptibility, are obtained by the WL algorithm over a broad range of temperature. It is discovered that for some realizations with system size larger than 8 3, sharp peaks are present in these physical quantities. These sharp peaks result from flipping a large domain, and furthermore, the critical domains are strongly correlated to the domains found at the zero temperature transition. This observation suggests a strong version of the zero temperature fixed point picture. The MKRG approximation is employed to test the correlation between the critical states and the ground states in much larger systems. It is found that the fraction of correlated realizations drops as system size grows, while there are still significant number of correlated realizations even for systems as large as 10243. Whether substantial correlations will remain in the thermodynamic limit is unclear from our present data.

•Jan 1, 2006•0 citations
The Ising model is a widely studied class of models in quantum computation. In this paper, we investigate the computational characteristics of the random-field Ising model (RFIM) with long-range interactions that decays as an inverse polynomial of distance, which can be achieved in the current ion trap system. We prove that for an RFIM with long-range interaction embedded on a two-dimensional plane, solving its ground state is NP-complete for all diminishing exponents, and prove that the one-dimensional RFIM with long-range interaction can be efficiently approximated when the interaction decays fast enough.
We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra{r}^{a}ra, where rrr is the distance between two lattice sites and a<0a<0a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.
Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.
In this thesis classical disordered spin systems, in particular, the random field Ising model (RFIM), are studied using numerical methods. The previous experimental and theoretical research on the RFIM is reviewed. Critical exponents of the RFIM are extracted at zero temperature. The heat capacity exponent α is found to be near zero. The ground states within a certain region with varying external field and disorder are portrayed, and it is speculated that if related quantities are scaled properly, ground state portraits at different system sizes are statistically similar. The performance of the Wang-Landau (WL) algorithm, which is used in our positive temperature study of the RFIM, is measured by the mean first-passage time. It is proved that the mean first, passage time of the mean field Ising model scales as N 2 log N, where N is the number of spins. The logarithmic correction term to the unbiased Markovian random walk is due to the boundaries. When applied to the Ising model with nearest neighbor interactions, the WL algorithm suffers from power law slowing down. Carefully designed cluster updating, however, can accelerate the WL algorithm and eliminate the slowing down. The WL algorithm is used in our finite temperature study of the RFIM. Physical quantities, such as the specific heat and the susceptibility, are obtained by the WL algorithm over a broad range of temperature. It is discovered that for some realizations with system size larger than 8 3, sharp peaks are present in these physical quantities. These sharp peaks result from flipping a large domain, and furthermore, the critical domains are strongly correlated to the domains found at the zero temperature transition. This observation suggests a strong version of the zero temperature fixed point picture. The MKRG approximation is employed to test the correlation between the critical states and the ground states in much larger systems. It is found that the fraction of correlated realizations drops as system size grows, while there are still significant number of correlated realizations even for systems as large as 10243. Whether substantial correlations will remain in the thermodynamic limit is unclear from our present data.